p-adic L-functions

The following is more a long comment than an answer per se.

One thing to keep in mind when discussing $p$-adic $L$-functions is that to a given algebraic automorphic representation $pi$ or Galois representation $rho$ is potentially attached several objects which could reasonably called the $p$-adic $L$-function of $rho/pi$. Largely for historical reasons, when one speaks of the $p$-adic $L$-function of $rho$ without further comment, one generally speaks of the $p$-adic $L$-function coming from the cyclotomic $mathbb Z_{p}$-extension, as I assume you do in your question. The most natural object from a strictly mathematical point of view seems to me to be the $p$-adic $L$-function attached to the universal deformation ring of $bar{rho}$ (at least when this universal deformation ring exists).

Even restricting yourself to the simplest case of the cyclotomic $p$-adic $L$-function, the case of $GL_{n}$ over $mathbb Q$ has not been done (that I know of) and I doubt (euphemism) that it will follow from the work of Eischen, Emerton, Harris, Li and Skinner (Emerton claims nothing of the sort). Unless I am very much mistaken, the cyclotomic case for $GL_{n}$ over $mathbb Q$ would be an extremely impressive progress. Somehow, the case of the anticyclotomic $mathbb Z_{p}$-extension of a CM field is sometimes easier because one can use the Rankin-Selberg method to prove that special values are algebraic and the Rankin-Selberg method is quite amenable to $p$-adic methods. I imagine that this is an ingredient in the work of EEHLS (but I know nothing about it, so please M.Emerton correct me if I'm wrong).

Leaving the real world for a second: conjecturally, cyclotomic $p$-adic $L$-functions are now constructed for any motive over $mathbb Q$ (though you will have a really hard time finding this in the literature, as one has to combine an impressive series of very involved papers). Of course, the conjectural construction would not tell you much in way of an actual construction (the conjectural construction gives you an element in some local cohomology group and you will have somehow to identify it as a global element), even though I admit I have been more than mildly impressed by an answer of Idoneal to a question here on MO about $p$-adic $L$-functions here which seems to indicate that analytic argument allows you to do just that in the case of modular forms.

Kevin Buzzard, sure the anticyclotomic $p$-adic $L$-function of an elliptic curve is part (technically, a specialization) of a two-variable $p$-adic $L$-function. In this setting and at least in the ordinary case, this has been known for more than 25 years (it was done in his thesis by S.Haran and later widely expanded by H.Hida in his Invent. Math. 79 paper). And further, this two-variable $p$-adic $L$-function is a specialization of a three-variable $p$-adic $L$-function taking into account variation of the weight in the Hida family passing through this elliptic curve. Even in the finite slope non-ordinary case, I think this three-variable $p$-adic $L$-function is known to exist by the work of A.Panciskin.